R.: On Superselection Rules in Bohm–Bell Theories

The meaning of superselection rules in Bohm–Bell theories (i.e., quantum the-ories with particle trajectories) is different from that in orthodox quantum the-ory. More precisely, there are two concepts of superselection rule, a weak and a strong one. Weak superselection rules exist both in orthodox quantum theory and in Bohm–Bell theories and represent the conventional understanding of super-selection rules. We introduce the concept of strong superselection rule, which does not exist in orthodox quantum theory. It relies on the clear ontology of Bohm– Bell theories and is a sharper and, in the Bohm–Bell context, more fundamental notion. A strong superselection rule for the observable G asserts that one can replace every state vector by a suitable statistical mixture of eigenvectors of G without changing the particle trajectories or their probabilities. A weak super-selection rule asserts that every state vector is empirically indistinguishable from a suitable statistical mixture of eigenvectors of G. We establish conditions on G for both kinds of superselection. For comparison, we also consider both kinds of superselection in theories of spontaneous wave function collapse. Key words: weak and strong superselection rules; Bohmian mechanics; Bell-type quantum field theory; beables; number operators; Ghirardi–Rimini–Weber model of spontaneous wave function collapse.